1. **State the problem:** Solve the equation $\tan x - \cot x = 0$ for $x=0$. 2. **Recall the definitions:** - $\tan x = \frac{\sin x}{\cos x}$ - $\cot x = \frac{\cos x}{\sin x}$ 3. **Rewrite the equation:** $$\tan x - \cot x = 0 \implies \tan x = \cot x$$ 4. **Express in terms of sine and cosine:** $$\frac{\sin x}{\cos x} = \frac{\cos x}{\sin x}$$ 5. **Cross-multiply:** $$\sin^2 x = \cos^2 x$$ 6. **Use the Pythagorean identity:** $$\sin^2 x - \cos^2 x = 0$$ 7. **Rewrite as:** $$\sin^2 x = \cos^2 x$$ 8. **Divide both sides by $\cos^2 x$ (assuming $\cos x \neq 0$):** $$\frac{\sin^2 x}{\cos^2 x} = \frac{\cos^2 x}{\cos^2 x} \implies \tan^2 x = 1$$ 9. **Take square root:** $$\tan x = \pm 1$$ 10. **Find general solutions:** $$x = \frac{\pi}{4} + k\pi \quad \text{or} \quad x = \frac{3\pi}{4} + k\pi, \quad k \in \mathbb{Z}$$ 11. **Check the specific value $x=0$:** $$\tan 0 = 0, \quad \cot 0 = \text{undefined}$$ So $x=0$ is not a solution because $\cot 0$ is undefined. **Final answer:** The solutions to $\tan x - \cot x = 0$ are $$x = \frac{\pi}{4} + k\pi \quad \text{or} \quad x = \frac{3\pi}{4} + k\pi, \quad k \in \mathbb{Z}$$ and $x=0$ is not a solution.